#223232

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(6,8)(9,10,13,17,14,18,11,15)(12,20,19,24,22,16,21,23)(25,26)(27,34)(28,32)(29,37)(30,33)(31,36)(35,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,18,17)(12,21,22,19)(16,24,20,23)(27,35)(31,39)(34,40)(36,38);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,27)(26,31)(28,36)(29,38)(30,35)(32,40)(33,39)(34,37);(2,5)(6,8)(9,11,14,13)(10,17,18,15)(12,19,22,21)(16,24,20,23)(25,28,30,29)(26,32,33,37)(27,36,35,38)(31,40,39,34)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=e,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130428);